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Tail behavior of the product of two dependent random variables with applications to risk theory

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Abstract

Let the random vector (X,Y) follow a bivariate Sarmanov distribution, where X is real-valued and Y is nonnegative. In this paper we investigate the impact of such a dependence structure between X and Y on the tail behavior of their product Z = XY. When X has a regularly varying tail, we establish an asymptotic formula, which extends Breiman’s theorem. Based on the obtained result, we consider a discrete-time insurance risk model with dependent insurance and financial risks, and derive the asymptotic and uniformly asymptotic behavior for the (in)finite-time ruin probabilities.

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References

  • Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  • Breiman, L.: On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10, 323–331 (1965)

    Article  MathSciNet  Google Scholar 

  • Chen, Y.: The finite-time ruin probability with dependent insurance and financial risks. J. Appl. Probab. 48, 1035–1048 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Y., Yuen, K.C.: Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stoch. Models 25, 76–89 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng, D., Ni, F., Pakes, A.G., Wang, Y.: Some properties of the exponential distribution class with applications to risk theory. J. Korean Stat. Soc. (2012). doi:10.1016/j.jkss.2012.03.002

    Google Scholar 

  • Cline, D.B.H., Samorodnitsky, G.: Subexponentiality of the product of independent random variables. Stoch. Process. Their Appl. 49, 75–98 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Denisov, D., Zwart, B.: On a theorem of Breiman and a class of random difference equations. J. Appl. Probab. 44, 1031–1046 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Falk, M., Hüsler, J., Reiss R.-D.: Laws of Small Numbers: Extremes and Rare Events. DMV Seminar 23, 3rd edn. Birkhäuser, Basel (2010)

    Google Scholar 

  • Gao, Q., Wang, Y.: Randomly weighted sums with dominated varying-tailed increments and application to risk theory. J. Korean Stat. Soc. 39, 305–314 (2010)

    Article  Google Scholar 

  • Goovaerts, M.J., Kaas, R., Laeven, R.J.A., Tang, Q., Vernic, R.: The tail probability of discounted sums of Pareto-like losses in insurance. Scand. Actuar. J. 6, 446–461 (2005)

    Article  MathSciNet  Google Scholar 

  • Hashorva, E.: Conditional limiting distribution of beta-independent random vectors. J. Multivar. Anal. 99, 1438–1459 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Hashorva, E.: Conditional limit results for type I polar distributions. Extremes 12, 239–263 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Hashorva, E.: Asymptotics of the norm of elliptical random vectors. J. Multivar. Anal. 11, 368–385 (2010)

    MathSciNet  Google Scholar 

  • Hashorva, E., Hüsler, J.: Extreme values in FGM random sequences. J. Multivar. Anal. 68, 212–225 (1999)

    Article  MATH  Google Scholar 

  • Hashorva, E., Pakes, A.G.: Tail asymptotics under beta random scaling. J. Math. Anal. Appl. 372, 496–514 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Hashorva, E., Pakes, A.G., Tang, Q.: Asymptotics of random contractions. Insurance Math. Econom. 47, 405–414 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Jessen, A.H., Mikosch, T.: Regularly varying functions. Institut Mathématique. Publications. Nouvelle Série. Publ. Inst. Math. (Beograd) 80, 171–192 (2006)

    Article  MathSciNet  Google Scholar 

  • Jiang, J., Tang, Q.: The product of two dependent random variables with regularly varying or rapidly varying tails. Stat. Probab. Lett. 81, 957–961 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Kotz, S., Balakrishnan, N., Johnson, N.L.: Continuous Multivariate Distributions, vol. 1: Models and Applications. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  • Lee, M.T.: Properties and applications of the Sarmanov family of bivariate distributions. Commun. Stat., Theory Methods 25, 1207–1222 (1996)

    Article  MATH  Google Scholar 

  • Liu, Y., Tang, Q.: The subexponential product convolution of two Weibulltype distributions. J. Aust. Math. Soc. 89, 277–288 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Maulik, K., Resnick, S., Rootzén, H.: Asymptotic independence and a network traffic model. J. Appl. Probab. 39, 671–699 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Norberg, R.: Ruin problems with assets and liabilities of difusion type. Stoch. Process. Their Appl. 81, 255–269 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Nyrhinen, H.: On the ruin probabilities in a general economic environment. Stoch. Process. Their Appl. 83, 319–330 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Nyrhinen, H.: Finite and in.nite time ruin probabilities in a stochastic economic environment. Stoch. Process. Their Appl. 92, 265–285 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Resnick, S.I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007)

    MATH  Google Scholar 

  • Rogozin, B.A., Sgibnev, M.S.: Banach algebras ofmeasures on the line with given asymptotics of distributions at infinity. Sib. Math. J. 40, 565–576 (1999)

    Article  MathSciNet  Google Scholar 

  • Shen, X., Lin, Z., Zhang, Y.: Uniform estimate for maximum of randomly weighted sums with applications to ruin theory. Methodol. Comput. Appl. Probab. 11, 669–685 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Tang, Q.: The subexponentiality of products revisited. Extremes 9, 231–241 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Tang, Q.: From light tails to heavy tails through multiplier. Extremes 11, 379–391 (2008a)

    Article  MathSciNet  MATH  Google Scholar 

  • Tang, Q.: Insensitivity to negative dependence of asymptotic tail probabilities of sums and maxima of sums. Stoch. Anal. Appl. 26, 435–450 (2008b)

    Article  MathSciNet  MATH  Google Scholar 

  • Tang, Q., Tsitsiashvili, G.: Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Process. Their Appl. 108, 299–325 (2003)

    MathSciNet  MATH  Google Scholar 

  • Tang, Q., Tsitsiashvili, G.: Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Probab. 36, 1278–1299 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Tang, Q., Vernic, R., Yuan, Z.: The finite-time ruin probability in the presence of dependent extremal insurance and financial risks. Preprint (2011)

  • Wang, D., Su, C., Zeng, Y.: Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory. Sci. China, Ser. A 48, 1379–1394 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, D., Tang, Q.: Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22, 253–272 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, K., Wang Y., Gao Q.: Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. (2011). doi:10.1007/s11009-011-9226-y

    Google Scholar 

  • Yi, L., Chen, Y., Su, C.: Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation. J. Math. Anal. Appl. 376, 365–372 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, Y., Shen, X., Weng, C.: Approximation of the tail probability of randomly weighted sums and applications. Stoch. Process. Their Appl. 119, 655–675 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou, M., Wang, K., Wang, Y.: Estimates for the finite-time ruin probability with insurance and financial risks. Acta Math. Appl. Sin., Engl. Ser. (2012). doi:10.1007/s10255-012-0189-8

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing Audit University, Nanjing, 210029, China

    Yang Yang

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Yang Yang

  3. Department of Mathematics, Soochow University, Suzhou, 215006, China

    Yuebao Wang

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  1. Yang Yang
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  2. Yuebao Wang
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Correspondence to Yuebao Wang.

Additional information

Research supported by National Natural Science Foundation of China (No. 11071182, 11001052), China Postdoctoral Science Foundation (No. 20100471365), Natural Science Foundation of Jiangsu Province of China (No. BK2010480), Postdoctoral Research Program of Jiangsu Province of China (No. 0901029C), Jiangsu Government Scholarship for Overseas Studies, Qing Lan Project.

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Yang, Y., Wang, Y. Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes 16, 55–74 (2013). https://doi.org/10.1007/s10687-012-0153-2

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  • DOI: https://doi.org/10.1007/s10687-012-0153-2

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